Exercise 7.  Consider the mapping  ,  where   denotes the principal branch of the square root function.  

If  ,  show that the lines    are mapped onto orthogonal curves.

Solution 7.

See text and/or instructor's solution manual.

Solution Method I.   Applying Theorem 10.1,       ,  

so that  ,  hence    is conformal at  .  

The lines    and    intersect orthogonally at the point  ,

therefore their image curves will intersect orthogonally at the point  .

Solution Method II.   In Example 2.13 in Section 2.2, we saw that the image curves are portions of hyperbolas.

Let  .  The vertical line   is mapped the right branch of the hyperbola  .  

Implicit differentiation of   produces the equation   ,  and then  .

The slope at the point    is   

                    .  

Let  .  The horizontal line   is mapped the upper branch of the hyperbola  .

Implicit differentiation of   produces the equation   ,  and then  .  

The slope at the point    is   

                    .  

Since  ,  the curves are orthogonal.

We are done.   

Aside.  If necessary, one can use the formula   in Exercise 4 and write

the slopes   at the point    as follows    

                    

and
                    .  

Since  ,  the curves are orthogonal.

We are really done.   

Aside.  We can let Mathematica illustrate our work.

                                   The transformation      

                                   The transformation      

We are really really done.

Aside.  We can let Mathematica investigate what happens at  .

                      At    the mapping   reduces angles by the factor  .

Remark.  In Section 2.2 we introduced the mapping  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell